Q: A compound bow stores elastic potential energy when drawn. If the area under the force vs. draw distance curve is $$11\ \mathrm{J}$$ and the arrow has mass $$0.026\ \mathrm{kg}$$, what is the maximum speed the arrow could ideally reach?

$$v = \sqrt{\frac{2 \times 11}{0.026}}$$

Q: If a spring is compressed by $$x$$ and then released, what is the total mechanical energy of the system (assuming no friction)?

$$E_{\text{total}} = \frac{1}{2}kx^2$

Q: A spring with $$k = 100\ \mathrm{N/m} is $$stretched by $$0.2\ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

$$U_s = \frac{1}{2} \times 100 \times (0.2)^2$ = 2\ \mathrm{J}$$

Question 1

A block of mass $$m is $$pressed against a spring with spring constant $$k$$, compressing it by a distance $$D. $$The block is released from rest and moves on a frictionless surface. What is the block's speed $$v$$ when it leaves the spring?

Accepted Answer

$$v = \sqrt{\frac{kD^2$}{m}}$$

Question 2

Which equation correctly represents the elastic potential energy stored in a spring compressed by a distance $$x$$?

Accepted Answer

$$U_s = \frac{1}{2}kx^2$

Question 3

A mass $$m is $$attached to a vertical spring with spring constant $$k$$ and gently lowered until it comes to rest at equilibrium, a distance $$D$$ below its initial position. What is the gravitational potential energy of the mass at this new position, relative to its starting point?

Accepted Answer

$$U_g = mg(D)$$

Question 4

A compound bow stores elastic potential energy when drawn. If the area under the force vs. draw distance curve is $$11\ \mathrm{J}$$ and the arrow has mass $$0.026\ \mathrm{kg}$$, what is the maximum speed the arrow could ideally reach?

Accepted Answer

$$v = \sqrt{\frac{2 	imes 11}{0.026}}$$

Question 5

Which of the following best describes the relationship between force and potential energy as a function of position $$x$$?

Accepted Answer

$$F(x) = -\frac{dU}{dx}$$

Question 6

On a potential energy graph, at which point is the force on the object zero?

Accepted Answer

Where $$\frac{dU}{dx} = 0$$

Question 7

A diatomic molecule's potential energy as a function of separation has a minimum at the equilibrium bond length. What does this minimum represent?

Accepted Answer

The most stable configuration of the molecule

Question 8

If a spring is compressed by $$x$$ and then released, what is the total mechanical energy of the system (assuming no friction)?

Accepted Answer

$$E_{	ext{total}} = \frac{1}{2}kx^2$

Question 9

Which statement about gravitational potential energy is correct?

Accepted Answer

It increases linearly with height: $$U_g = mgh$$

Question 10

A block is released from rest after compressing a spring by $$D. If $$the block has mass $$m$$ and the spring constant is $$k$$, what is the minimum compression $$D$$ required for the block to reach a speed $$v$$?

Accepted Answer

$$D = v \sqrt{\frac{m}{k}}$$

Question 11

When analyzing energy changes in a system, which of the following is NOT a type of energy considered in mechanical energy?

Accepted Answer

Thermal energy due to friction

Question 12

If the potential energy curve of a system has a local minimum at $$x_0$$, what type of equilibrium does this point represent?

Accepted Answer

Stable equilibrium

Question 13

A spring with $$k = 100\ \mathrm{N/m} is $$stretched by $$0.2\ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

Accepted Answer

$$U_s = \frac{1}{2} 	imes 100 	imes (0.2)^2$ = 2\ \mathrm{J}$$

Question 14

Which of the following best explains why energy must be added to break a molecular bond?

Accepted Answer

The atoms are at a potential energy minimum and require energy to separate

Question 15

On a potential energy graph, where is the object's speed the largest?

Accepted Answer

Where the potential energy is lowest

Question 16

If a hand gently lowers a mass attached to a vertical spring until it comes to rest, what work is done by the hand?

Accepted Answer

Negative work equal in magnitude to the increase in spring potential energy

Question 17

A block is moving on a ramp with a potential energy graph shown. At which point is the block in unstable equilibrium?

Accepted Answer

At a local maximum of the potential energy curve

Question 18

Which of the following statements about energy in ATP hydrolysis is correct?

Accepted Answer

Energy is released because breaking the bond moves the system to a lower potential energy

Question 19

If the force on an object is positive, what can be said about the slope of the potential energy curve at that position?

Accepted Answer

The slope is negative: $$\frac{dU}{dx} \u003c 0$$

Potential Energy in Mechanical Systems: Springs, Gravity, and Energy Graphs

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